论文标题
Wigner合奏的日志决定因素的边缘CLT
An edge CLT for the log determinant of Wigner ensembles
论文作者
论文摘要
我们为$ \ log \ left \ vert \ det \ left（w_ {n} -e_ {n} \ right）\ right \ vert，$ w_ {n} $是wigner matrix，$ e_ {n} $是semi-circ的边缘。准确地说，$ e_n = 2+n^{ - 2/3}σ_n$，$σ_n$是常数（可能为负），或者是正实数的序列，慢慢散布到无穷大，因此$σ_n\ ll \ ll \ log^log^{2} n $。我们还扩展了CLT以覆盖尖刺的Wigner矩阵。我们对CLT的兴趣是由于其在批判性尖刺模型中的统计测试以及球形Sherrington-Kirkpatrick统计物理模型中的自由能的波动而进行的。
We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}σ_N$ with $σ_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $σ_N \ll \log^{2} N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spherical Sherrington-Kirkpatrick model of statistical physics.
